# NCERT Solutions for Class 9 Maths Chapter 10 Circles

NCERT Solutions for Class 9 Maths Chapter 10 Circles: Many objects that we come across in our daily life is circular in shape, such as ring, bangle, wheels of the vehicle, clock, etc. So these NCERT Solutions for Class 9 Maths Chapter will help in solving many problems related to circular shapes. A circle is obtained by connecting all such points which are at a fixed distance from a fixed point. The fixed distance is called the radius of the circle and the fixed point is called the centre of the circle. A circle divides the plane in which the circle lies into three parts as shown in figure 1, which are the interior of the circle, the exterior of the circle and the circle. The yellow shaded region in figure1 is the interior of the circle and the black border is the circle and the area outside the circle is the exterior of the circle. The NCERT Solutions for Class 9 Maths Chapter 10 deals with the circle and some of its properties.

Figure 1

NCERT Solutions for Class 9 of this chapter explains about chords, segments, and sectors

Chord: Suppose if we locate two points on a circle say P and Q as shown in figure 2, then the line joining P and Q are known as the chord of the circle

Figure 2

The line PQ is the chord of the circle. The largest chord of a circle passes through the centre of the circle and is known as diameter. In figure 2, AB is the diameter and the length of the diameter is twice the radius. That is AB= 2 OA or 2 OB. Here OA and OB are radii of the circle.

Arc: The point PQ divides the circle into two parts, the major arc PQ and minor arc PQ ( Figure 3).

Figure 3

Segment: The region between the chord and either of its arc is called the segment. The minor and major segment is represented by figure 4

Figure 4

Sector: The region between an arc and the two radii, joining the centre to the endpoints of the arc is called a sector. The major and minor sectors are represented in figure 5. OP=OQ=radius of the circle

Figure 5

Certain properties of circles mentioned in the NCERT Solutions of this chapter are given below:

1. Equal chords of a circle subtend equal angle at the centre of the circle. That is in figure 6 AB and CD are equal chords and O is the centre. Then angle BOA=angle DOC.

Figure 6

2. If the angle subtended by chords at the centre of a circle are equal then the chords are equal. This is the converse of property 1.

3. The perpendicular from the centre of the circle to a chord bisects the chord. That is OM is perpendicular to the chord AB then AM=BM (figure 7)

Figure 7

4. If the line is drawn from the centre of a circle bisects a chord then the line is perpendicular to the chord. This property is vice-versa of  property 3

5. Equal chords of a circle are equidistant from the centre. That is in figure 8 PQ=RS then distance OL=OM

6. Chords equidistant from the centre are equal in length. That is if the perpendicular distance OL and OM are equal then PQ=RS (figure 8)

7. The angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle.

There are a few more properties of cyclic quadrilaterals mentioned in the NCERT Solutions for Class 9 Maths Chapter 10.

Figure 9

8. The sum of either pair of opposite angles of a cyclic quadrilateral is 180 degrees. That is in figure 9

$< ADC+< ABC=180^0\ and\ < BCD+< BAD=180^0$

9. If the sum of a pair of opposite angles of a quadrilateral is 180 degrees, the quadrilateral is cyclic.

NCERT Solutions for Class 9 Maths Chapter 10 Circles consist of six exercises. All the questions are mandatory in exam point of view.

## NCERT Solutions for Class 9 Maths - Chapter Wise

 Chapter No. Chapter Name Chapter 1 Number Systems Chapter 2 Polynomials Chapter 3 Coordinate Geometry Chapter 4 Linear Equations In Two Variables Chapter 5 Introduction to Euclid's Geometry Chapter 6 Lines And Angles Chapter 7 Triangles Chapter 8 Quadrilaterals Chapter 9 Areas of Parallelograms and Triangles Chapter 11 Constructions Chapter 12 Heron’s Formula Chapter 13 Surface Areas and Volumes Chapter 14 Statistics Chapter 15 Probability